When parameters are tied across multiple layers in a neural network, they share the exact same underlying tensor. During backpropagation, the gradients computed for each specific instance of the shared layer are summed together. Because the single parameter tensor must account for the error propagated through all the distinct layers where it was applied during the forward pass, these individual layer gradients are added to correctly determine the final gradient for the tied parameters.

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In deep learning frameworks, parameters can be shared or 'tied' across multiple layers within a sequential model. This is typically achieved by explicitly allocating a layer—such as a fully connected layer—and referencing that exact same object multiple times when constructing the network architecture. Because the tied layers are represented by the identical underlying parameter tensor in memory, they are not merely equal in value; any modification applied to the parameters of one layer will immediately alter the parameters of all other layers tied to it.

Tied Parameters in Sequential Models

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Gradients of Tied Parameters

In TensorFlow's Keras API, attempting to tie parameters by adding the exact same layer object multiple times to a `Sequential` model behaves differently than in other deep learning frameworks. Instead of reusing the layer at multiple steps, `tf.keras.models.Sequential` automatically removes any duplicate layer instances from the network architecture. Consequently, to achieve parameter sharing in TensorFlow without altering the sequence of layers, developers typically rely on the Functional API or model subclassing rather than the `Sequential` class.

Duplicate Layer Removal in TensorFlow Sequential Models

In MXNet, parameter sharing can be implemented explicitly by initializing a new layer with the parameters of an existing layer. When constructing the network, passing the parameters of an existing layer via the `params` argument (e.g., `nn.Dense(..., params=shared.params)`) to a newly instantiated layer ensures that both layers utilize the exact same underlying parameter tensor, effectively tying their parameters together while remaining distinct layer objects.

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